Roger Allaway is your man for any questions on this sort of thing. He's one of the leading experts on American soccer history, particularly of this sort of era. I'd drop him a PM to let him know about the thread.
I've done a little research on the subject, and it seems to me that the largest group was a bunch of pretty average Scottish and English players who moved to the United States either because pay in the UK was dismally low, or they were semipro players who immigrated to the US and then made successful careers in the US. However, probably the single largest contingent of notables came from the Hakoah All-Stars, an all Jewish team that toured the US in 1926. These included Erno Schwarz, Moritz Haeusler, Bela Guttmann, Laszlo Sternberg, and József Eisenhoffer. Several of these people went on to quite distinguished careers as managers. Some other European "All-Stars": Harold Brittan (Chelsea), Tommy Muirhead (Rangers), Tec White (Motherwell), Sam Chedgzoy (Everton, England), Alex Hunter (Tottenham, Wigan), Kálmán Konrád, Alex McNab (Greenock Morton), György Molnár, Janos Nehadoma, and Gerrit Visser (Holland). A good number of other players played a low-level soccer in Europe, moved to the US for non-sporting reasons, and ended up establishing themselves as pretty good players. The most important of these was probably Barney Battles Jr., who starred for Hearts and Scotland for quite a while after turning pro in the US. Others include Andy Auld, Tommy Fleming, George Moorhouse, Bobby Yule, and Werner Nilsen.
Nice quote ChrisE Excluding Alfred Schaffer, the most impressive names are Bela Guttmann & Kalman Konrad. Arguably top-3 material in their respective roles by that time.
Thanks. I assume you're referring to Guttman and Konrad's roles as players? They're probably the two players with the most distinguished coaching careers listed. I'm really not familiar with European soccer history in the 20's, so I'm no judge of quality, but I'd say that Sam Chedgzoy and Tommy Muirhead had pretty significant careers in Europe as well. Chedgzoy was capped 8 times by England, and won the English First Division with Everton in 1915. Muirhead was capped 13 times, and had a very lengthy career with Rangers. Apart from the Wikipedia page, I can't find any reference to Albert Schaffer playing in the United States. His Hungarian page also makes no mention of it, although there's a gap between 1926 and 1933. Presumably he played here after the ASL sort of collapsed in 1931? Does anybody (Alex_K) have any further information about his time in the United States?